Given `Delta=|(x,x^2,1+px^3),(y,y^2,1+py^3),(z,z^2,1+pz^3)|` Express this as sum of two determinants.

Given Delta=|(x,x^2,1+px^3),(y,y^2,1+py^3),(z,z^2,1+pz^3)| Prove that Delta=(1+pxyz)(x-y)(y-z)(z-x) .

Consider the determinant Delta=abs[[x,x^2,1+x^3],[y,y^2,1+y^3],[z,z^2,1+z^3]] , Where x,y,z are different. Express the above determinant as sum of two determinants.

Consider the determinant Delta=abs[[x,x^2,1+x^3],[y,y^2,1+y^3],[z,z^2,1+z^3]] , Where x,y,z are different. Show that if Delta=0 ,then 1+xyz=0

Prove that |(1,x,x^3),(1,y,y^3),(1,z,z^3)|=(x+y+z)(x-y)(y-z)(z-y) .

Given Delta=|(y+z,x,y),(z+x,z,x),(x+y,y,z)| Prove that Delta=(x+y+z)(x-z)^2 .

Given Delta=|(y+z,x,y),(z+x,z,x),(x+y,y,z)| Apply R_1rarrR_1+R_2+R_3 .

Let |A|=|[x_1, x_2, x_3],[ y_1, y_2, y_3],[ z_1 , z_2, z_3]| .Expand |A| along the first row and along the first column. Comment on the two results.

Given Delta=|(y+z,x,y),(z+x,z,x),(x+y,y,z)| Take (x+y+z) common from R_1 .

Prove that abs[[1,x,x^2],[1,y,y^2],[1,z,z^2]]=(x-y)(y-z)(z-x)

Delta_(1) = |(y^(5)z^(6) (z^(3)-y^(3)),x^(4)z^(6)(x^(3)-z^(3)),x^(4)y^(5)(y^(3)-x^(3))),(y^(2)z^(3)(y^(6)-z^(6)),xz^(3)(z^(6)-x^(6)),xy^(2)(x^(6)-y^(6))),(y^(2)z^(3)(z^(3)-y^(3)),xz^(3)(x^(3)-z^(3)),xy^(2)(y^(3)-x^(3)))| and Delta_(2)=|(x,y^(2),z^(3)),(x^(4),y^(5),z^(6)),(x^(7), y^(8),z^(9))| Then Delta_(1) Delta_(2) is equal to a) Delta_(2)^(2) b) Delta_(2)^(3) c) Delta_(2)^(4) d)None of these